An exact witness and certificate for a recorded negative answer

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For the preceding design question, $V(f_1,\ldots,f_r)$ is the common zero set of the constructed interpolation polynomials, while $D$ is the finite design. A negative answer means those polynomials can vanish at points outside $D$, despite belonging to the design ideal.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. This duplicates the status context of canonical record 20001223. The source gives no counterexample coordinates or proof in this sentence; it is preserved as a historical source remark. The supplied catalogue marks this record open; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

Question 47 was answered in the negative by Bernd Sturmfels at the conference. [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

This record reports a negative answer to Question 47 rather than asking another question.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001224 (AIM-COMPUTATION-0062). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-COMPUTATION-0062. https://aimath.org/WWN/compalgstat/compalgstat.pdf.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20001224 and code AIM-COMPUTATION-0062 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 20001223. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The primary source's sentence following Question 47 reports its negative resolution; it is not a separate conjecture. The catalogue open label is retained as supplied, without interpreting it as evidence that this historical remark poses an open problem. A witness and an independently checkable repair are investigated below.

Research attempt: Verifying the recorded counterexample through an exact repair certificate

For a standalone check take $D=\{(0,0),(1,1),(2,2),(0,1)\}$ and $\mathcal O=\{1,x,y,xy\}$. Evaluation on $D$ is injective on $\operatorname{span}\mathcal O$: a vanishing $a+bx+cy+dxy$ gives $a=0$, $b+c+d=0$, $2b+2c+4d=0$, and $c=0$, hence all coefficients vanish. The four monomials are the standard monomials of $(x^2,y^2)$. Their interpolation of its generators yields \[ f_1=x^2-xy,\qquad f_2=y^2-xy+x-y. \] Both vanish everywhere on $x=y$, so the common zero $(3,3)$ verifies the reported negative answer.

To test a stronger replacement for the failed corner-generation route, set $h=xy^2-3xy+2x$ and $J=(f_1,f_2,h)$. Here one can prove the exact ideal identity $J=I(D)$, not only equality of zero sets. In the quotient by $J$ the relations are \[ x^2=xy,\quad y^2=xy-x+y,\quad xy^2=3xy-2x,\quad x^2y=xy^2=3xy-2x. \] The last identity follows by multiplying $f_1$ by $y$. Multiplication by either variable therefore preserves the span of the four residue classes of $\mathcal O$: products of $1,x,y$ reduce by the first two relations, and products of $xy$ reduce by the last two. Since that span contains $1$, induction on total degree shows that every monomial, hence every polynomial, reduces to it. Thus $\dim_{\mathbb Q}\mathbb Q[x,y]/J\le4$.

All three generators vanish on $D$, so evaluation descends from this quotient to $\mathbb Q^D$. The four classes are independent under evaluation by the earlier calculation. Every residue class consequently has a unique representative in $\operatorname{span}\mathcal O$, and a residue class vanishing on all four points is zero. Thus $\ker(\mathrm{evaluation})=J=I(D)$, as claimed. The same proof works over $\mathbb C$, because the evaluation determinant remains nonzero. In particular the repaired quotient has no extra multiplicities: it is isomorphic as an algebra to the fourfold product of the coefficient field by evaluation.

This calculation pinpoints the missing information in the original corner test. Closing the proposed quotient basis under multiplication needs the degree-three relation $h$, which is absent from interpolation of the two minimal generators alone.

Verification and adversarial audit

Every reduction used in the dimension argument is displayed, including the relation for $x^2y$; no terminating monomial-order reduction is assumed. Surjectivity and injectivity of evaluation are obtained from the four independent classes, so the ideal identity does not rely on an unproved radicality assertion.

Audit corrections.

The open metadata and the source's recorded negative answer are both preserved and explicitly distinguished. An exact ideal certificate replaces an unsupported inference from equality of zero sets.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

This verifies a known negative answer and an elementary repair of one chosen witness. The original record is only a historical status sentence. No new general theorem, source example attribution, or present-day open status is claimed.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/compalgstat/compalgstat.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.